Write the terms of ∑i=1^4 f(xi) Δx, with x1=0, x2=2, x3=4, x4=6, and Δx=0.5, for each function.

step 1 All right, in this video, we're looking at evaluating a sigma. I'm using this particular format

Write the terms of ∑i=1^4 f(xi) Δx, with x1=0, x2=2, x3=4,...

Write the terms of $\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,$ with $x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,$ and $\Delta x=0.5,$ for each function. Evaluate the sum. See Example $5(c)$
$$f(x)=x^{2}-1$$

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Key Concepts

Riemann Sum

A Riemann sum approximates the area under a curve by dividing an interval into subintervals, evaluating the function at specified points within these subintervals, and summing the products of these function values and the subinterval width. This method is fundamental in understanding how integrals are approximated and defined.

Partitioning the Interval

Partitioning involves dividing the domain of the function into smaller, equally spaced segments, each of width ?x. These segments are used as the basis for evaluating the function and accumulating area approximations in a Riemann sum.

Function Evaluation

In a Riemann sum, the function is evaluated at specific chosen points (often the endpoints or midpoints of the subintervals) within the partition. These evaluations provide the heights of the rectangles or other shapes whose areas approximate the area under the curve.

Summation Notation

Summation notation (?) is used to succinctly express the accumulation of terms, such as the areas of subintervals in a Riemann sum. It indicates that computations are performed over a series of indexed terms that contribute to the overall sum.